Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

TL;DR

This paper investigates the mean curvature flow with surgery for submanifolds in spheres under quadratic curvature pinching conditions, leading to topological classifications of the submanifolds as spheres or connected sums of handles.

Contribution

It establishes the existence of a mean curvature flow with surgery under specific quadratic pinching conditions and classifies the topology of resulting submanifolds.

Findings

Flow with surgery exists under the pinching condition.

Submanifolds are diffeomorphic to spheres or connected sums of handles.

Results are sharp for dimensions n ≥ 8.

Abstract

We study mean curvature flow of $n$-dimensional submanifolds of $S_K^{n+\ell}$, the round $(n+\ell)$-sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A|^{2} < \frac{1}{n-2}|H|^{2} + 4K$ when $n\geq 8$, $|A|^{2} < \frac{4}{3n}|H|^{2}+\frac{n}{2}K$ when $n=7$, and $|A|^2<\frac{3(n+1)}{2n(n+2)}|H|^2+\frac{2n(n-1)}{3(n+1)}K$ when $n=5$ or $6$. This condition is related to a theorem of Li and Li [Arch. Math., 58:582--594, 1992] which states that the only $n$-dimensional minimal submanifolds of $S_K^{n+\ell}$ satisfying $|A|^2<\frac{2n}{3}K$ are the totally geodesic $n$-spheres. We prove the existence of a suitable mean curvature flow with surgeries starting from initial data satisfying the pinching condition. As a result, we conclude that any smoothly, properly immersed submanifold of $S_K^{n+1}$ satisfying the pinching condition is diffeomorphic either to the sphere $S^n$ or to the connected sum of a finite number of handles $S^1\times S^{n-1}$. The results are sharp when $n\geq 8$ due to hypersurface counterexamples.